Can I convert a heavily triangulated STL file to quads?
3D scanned files are not only heavily triangulated, but also tend to have loads of polygons. As with every question, it depends on the purpose of your final result (still image, animation, 3D game asset, printing) if this is good or bad. Another important but missing information would be the kind of model we're talking about (thus my answer will be quite broad). Things can work on a 3D scan of a rock but that doesn't mean that they also work on a 3D scan of a person. 

Blender offers some basic possibilities to convert triangles to quads (mainly the decimate modifier) but it only rarely does its job in a way that you could call the result a mesh with clean topology. 

There are also tools out there which are specifically programmed with the goal of creating clean topologized meshes. Instant Meshes is one of them.

As someone whose primarily focus is creating game assets, I would  go the "hard" way and do the re-topology by hand, as it's shown in numerous tutorials around the internet. You could then use your scanned model as the high-poly model and bake all the medium details as a displacement map and all the fine details as a normal map. These models can also be used in still renderings, animations and of course in 3D printing. 

Again, this answer is very broad due to the lack of information regarding the purpose and the form of the model in general.